Question: Solve the exponential equation for $x$. 64 x − 6 ⋅ 4 9 x − 3 = 64 6 x + 4 64\^{ x-6}\cdot 4\^{ 9x-3}=64\^{ 6x+4} $x=$
Answer: The strategy Let's write $4$ in base $64$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $64$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 64 x − 6 ⋅ 4 9 x − 3 = 64 x − 6 ⋅ ( 64 1 3 ) 9 x − 3 = 64 x − 6 ⋅ 64 3 x − 1 = 64 x − 6 + ( 3 x − 1 ) = 64 4 x − 7 ( 4 = 64 1 3 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 64\^{ x-6}\cdot 4\^{9x-3}&=64\^{ x-6}\cdot (64\^{ \frac13})\^{ 9x-3} &&&&(4=64\^{ \frac13})\\\\ &=64\^{C{x-6}}\cdot 64\^{ {3x-1}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=64\^{ C{x-6} \ + \ ({3x-1}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=64\^{ 4x-7} \end{aligned} Solving the linear equation We obtain the following equation. 64 4 x − 7 = 64 6 x + 4 64\^{ 4x-7}=64\^{ 6x+4} Now we can equate the exponents and solve for $x$. $\begin{aligned} 4x-7&=6x+4\\\\ x &= -\dfrac{11}{2}\end{aligned}$ The answer The answer is $x=-\dfrac{11}{2}$. You can check this answer by substituting $\it{x=-\dfrac{11}{2}}$ in the original equation and evaluating both sides.